A Gaussian Mersenne prime is a Gaussian prime of the form
 |
Here 
is a positive integer. The Gaussian
integer 
is called a Gaussian Mersenne number even when it is not a Gaussian
prime. The corresponding number 
is its complex conjugate.
In terms of the squared complex modulus,
 |
Consequently, 
is a Gaussian prime iff 
is an ordinary prime
(Berrizbeitia and Iskra 2010).
The known exponents 
are 2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163,
167, 239, 241, 283, 353, 367, 379, 457, 997, ... (OEIS A057429).
Other than 2, every such exponent must itself be a prime. The exponents satisfying
begin 73, 113, 241, 353, 457, 3041, 27529, 364289, 991961, 1203793, 1667321, 4792057,
.... They give perfect Gaussian integers
by McDaniel's construction (McDaniel 1974).
As of Aug. 2026, the largest Gaussian Mersenne prime listed by the Prime Pages has 
. Its squared complex
modulus, 
, is a prime
number containing 4,610,945 digits (Caldwell).
